Transcript 슬라이드 1 - Pusan

Graphs
2011, Fall
Pusan National University
Ki-Joune Li
PNU
STEM
Graph

Definition


G = (V,E )
where V : a set of vertices and
E = { <u ,v > | u ,v  V } : a set of edges
Some Properties


Equivalence of Graphs
Tree


a Graph
Minimal Graph
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a
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a
e
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Some Terms

Directed Graph



Complete Graph



G is a directed graph iff <u,v >  <u,v > for u  v  V
Otherwise G is an undirected graph
Suppose nv = number(V )
Undirected graph G is a complete graph if ne = nv (nv - 1) / 2,
where ne is the number of edges
Adjacency


For a graph G=(V,E )
u is adjacent to v iff  e = <u,v > E , where u,v  V
If G is undirected, v is adjacent to u
otherwise v is adjacent from u
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Some Terms

Subgraph


Path from u to v



G’ is a subgraph of a graph G iff
V (G’ )  V (G ) and E (G’ )  E (G )
A sequence of vertices u=v0, v1, v2, … vn=v  V , such that
<vi ,vi +1>  E for every vi
Cycle iff u = v
Connected



Two vertices u and v are connected iff  a path from u to v
Graph G is connected iff for every pair u and v there is a path
from u to v
Connected Components

A connected subgraph of G
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Some Terms

Connected



An directed graph G is strongly connected iff
iff for every pair u and v there is a path from u to v and
path from v to u
DAG: directed acyclic graph (DAG)
In-degree and Out-Degree


In-degree of v : number of edges coming to v
Out-degree of v : number of edges going from v
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Representation of Graphs: Matrix

Adjacency Matrix


A[i, j ] = 1, if there is an edge <vi ,vj >
A[i, j ] = 0, otherwise
0
1
Example
0 1 1 1
0 0 1 0
2
3
0 0 0 1
0 0 0 0
Undirected Graph: Symmetric Matrix
 Space complexity: O (nv2 ) bits
 In-degree (out-degree) of a node

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Representation of Graphs: List

Adjacency List

Each node has a list of adjacent nodes
0
1
1
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3
0
3


Space Complexity: O (nv + ne )
Inverse Adjacent List
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0
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0
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Weighted Graph: Network

For each edge <vi ,vj >, a weight is given


Example: Road Network
Adjacency Matrix


A[i, j ] = wij, if there is an edge <vi ,vj > and wij is the weight
A[i, j ] = , otherwise

1.5
2.3
1.2


1.0




1.9




Adjacency List
0
1 1.5
1
2 1.0
2
3 1.9
3
0
1.5
1
1.0
2.3
1.2
2
1.9
2 2.3
3
3 1.2
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Graph: Basic Operations

Traversal




Depth First Search (DFS)
Breadth First Search (BFS)
Used for search
Example

Find Yellow Node from 0
0
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DFS: Depth First Search
void Graph::DFS() {
visited=new Boolean[n];
for(i=0;i<n;i++)visited=FALSE;
DFS(0);
delete[] visited;
}
0
1
4
2
void Graph::DFS(int v) {
visited[v]=TRUE;
for each u adjacent to v {
if(visited[u]==FALSE) DFS(w);
}
}
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Adjacency List: Time Complexity: O(e)
Adjacency Matrix: Time Complexity: O(n2)
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BFS: Breadth First Search
void Graph::BFS() {
visited=new Boolean[n];
for(i=0;i<n;i++)visited=FALSE;
Queue *queue=new Queue;
queue.insert(v);
while(queue.empty()!=TRUE) {
v=queue.delete()
for every w adjacent to v) {
if(visited[w]==FALSE) {
queue.insert(w);
visited[w]=TRUE;
}
}
}
delete[] visited;
}
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1
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7
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Adjacency List: Time Complexity: O(e)
Adjacency Matrix: Time Complexity: O(n2)
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Spanning Tree

A subgraph T of G = (V,E ) is a spanning tree of G

iff T is tree and V (T )=V (G )
0
1
4

Finding Spanning Tree: Traversal


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DFS Spanning Tree
BFS Spanning Tree
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Articulation Point and Biconnected
Components

Articulation Point

A vertex v of a graph G is an articulation point iff
the deletion of v makes G two connected components
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5

Biconnected Graph


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6
3
Connected Graph without articulation point
Biconnected Components of a graph
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Finding Articulation Point: DFS Tree

Back Edge and Cross Edge of DFS Spanning Tree



Tree edge: edge of tree
Back edge: edge to an ancestor
Cross edge: neither tree edge nor back edge

No cross edge for DFS spanning tree
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0
Back edge
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0
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Back edge
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Finding Articulation Point

Articulation point


Root is an articulation point if it has at least two children.
u is an articulation point
if  child of u without back edge to an ancestor of u
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Back edge
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0
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Back edge
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Finding Articulation Point by DFS Number

DFS number of a vertex: dfn (v )

0
The visit number by DFS
1

Low number: low (v )


1
low (v )= min{
dfn (v),
min{dfn (x)| (v, x ): back edge },
min{low (x)| x : child of v } }
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2
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v is an articulation Point


If v is the root node with at least two children or
If v has a child u such that low (u)  dfn (v)

v is the boss of subtree: if v dies, the subtree looses the line
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Computation of dfs(v) and low(v)
void Graph::DfnLow(x) {
num=1; // num: class variable
for(i=0;i<n;i++) dfn[i]=low[i]=0;
DfnLow(x,-1);//x is the root node
}
void Graph::DfnLow(int u,v) {
dfn[u]=low[u]=num++;
for(each vertex w adjacent from u){
if(dfn[w]==0) // unvisited w
DfnLow(w,u);
low[w]=min(low[u],low[w]);
else if(w!=v)
low[u]=min(low[u],dfn[w]); //back edge
}
}
Adjacency List: Time Complexity: O(e)
Adjacency Matrix: Time Complexity: O(n2)
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Minimum Spanning Tree

G is a weighted graph

T is the MST of G iff T is a spanning tree of G and
for any other spanning tree of G, C (T ) < C (T’ )
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Finding MST

Greedy Algorithm


Choosing a next branch which looks like better than others.
Not always the optimal solution
Globally optimal solution
Solution by greedy algorithm,
only locally optimal
Current
state

Kruskal’s Algorithm and Prim’s Algorithm

Two greedy algorithms to find MST
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Kruskal’s Algorithm
Algorithm KruskalMST
Input: Graph G
Output: MST T
Begin
T {};
while( n(T)<n-1 and G.E is not empty) {
(v,w) smallest edge of G.E;
G.E  G.E-{(v,w)};
if no cycle in {(v,w)}T, T {(v,w)}T
}
if(n(T)<n-1) cout<<“No MST”;
End Algorithm
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7X
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10
T is MST
Time Complexity: O(e loge)
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Checking Cycles for Kruskal’s Algorithm
V
1
V1
V2
log n
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If v  V and
8
w  V,
then  cycle,
otherwise no cycle
w
v
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Prim’s Algorithm
Algorithm PrimMST
Input: Graph G
Output: MST T
Begin
Vnew {v0}; Enew {};
while( n(Enew)<n-1) {
select v such that (u,v) is the smallest edge
where u Vnew, v Vnew;
if no such v, break;
G.E  G.E-{(u,v)};
Enew {(u,v)} Enew;
Vnew {v}  Vnew;
}
if(n(T)<n-1) cout<<“No MST”;
End Algorithm
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T is the MST
Time Complexity: O( n
4
2)
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Finding the Edge with Min-Cost
Step 1
Step 2
T
V
TV
TV
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0
1
1
3
3
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2
n-1
n
n + (n-1) +… 1 = O( n2 )
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Shortest Path Problem

Shortest Path

From 0 to all other vertices
0
5
vertex
cost
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Finding Shortest Path from Single Source
(Nonnegative Weight)
2
0
Algorithm DijkstraShortestPath(G) /* G=(V,E) */
output: Shortest Path Length Array D[n]
Begin
S {v0};
D[v0]0;
for each v in V-{v0}, do D[v]  c(v0,v);
while S  V do
begin
choose a vertex w in V-S such that D[w] is minimum;
add w to S;
for each v in V-S do
D[v]  min(D[v],D[w]+c(w,v));
end
End Algorithm
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w
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O(
n2 )
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Finding Shortest Path from Single Source
(Nonnegative Weight)
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∞
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3: {v0, v1, v2}
2: {v0, v1}
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∞
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1: {v0}
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∞
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4: {v0, v1, v2, v4}
5: {v0, v1, v2, v3, v4}
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Transitive Closure

Transitive Closure

Set of reachable nodes
0
0
1
0
1
1
2
2
4
2
4
4
3
3
3
A+
A
A*
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1
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1
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1
0
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1
0
0
0
0
1
1
1
0
0
1
1
1
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Transitive Closure
Void Graph::TransitiveClosure {
for(int i=0;i<n;i++)
for (int j=0;i<n;j++)
a[i][j]=c[i][j];
for(int k=0;k<n;k++)
for(int i=0;i<n;i++)
for (int j=0;i<n;j++)
a[i][j]= a[i][j]||(a[i][k] && a[k][j]);
}
O ( n3 )
Void Graph::AllPairsShortestPath {
for(int i=0;i<n;i++)
for (int j=0;i<n;j++)
a[i][j]=c[i][j];
for(int k=0;k<n;k++)
for(int i=0;i<n;i++)
for (int j=0;i<n;j++)
a[i][j]= min(a[i][j],(a[i][k]+a[k][j]));
}
O ( n3 )
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Activity Networks
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AOV – Activity-on-vertex
Node 1 is a immediate successor of Node 0
0
Node 0 is a immediate predecessor of Node 1
1
Node 0  Node 1
4
2
Node 1  Node 5
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5
Node 5 is a successor of Node 0
7
6
Node 0 is a successor of Node 5
Node 0  Node 5
Partial order: for some pairs (v, w) (v, w  V ), v  w (but not for all pairs)
(cf. Total Order: for every pair (v, w) (v, w  V ), v  w or w  v )
No Cycle
Transitive and Irreflexive
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Topological Order
0
Ordering: (0, 1, 2, 3, 4, 5, 7, 6)
1
Ordering: (0, 1, 4, 2, 3, 5, 7, 6)
4
2
Both orderings satisfy the partial order
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5
7
6
Topological order: Ordering by partial order
Ordering: (0, 1, 2, 5, 4, 3, 7, 6): Not a topological order
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Topological Ordering
0
0
1
1
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7
0, 1
0, 1, 4
0, 1, 4, 3
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Topological Ordering
Void Topological_Ordering_AOV(Digraph G)
for each node v in G.V {
if v has no predecessor {
cout << v;
O( n + e )
delete v from G.V;
deleve (v,w) from G.E;
}
}
if G.V is not empty, cout <<“Cycle found\n”;
}
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AOE – Activity-on-edge
PERT (Project Evaluation and Review Technique
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3
1
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12
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11
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2
Node 2 is completed
only if every predecessor is completed
Earliest Time: the LONGEST PATH from node 0
 Earliest time of node 1: 5
 Earliest time of node 2: 8
 Earliest time of node 4: 8
 Earliest time of node 5: max(8+11, 5+2, 8+6)=19
 Earliest time of node 7: max(8+9, 19+4)=23
 Earliest time of node 6: max(23+12, 19+6)=35
5
3
1
(0, 1, 4, 5, 7, 6) : Critical Path
By reducing the length on the critical path,
we can reduce the length of total path.
9
4
8
1
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11
0
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2
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12
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